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Thesis On A Computational Study on Entropy Generation Minimization of Pipe Bending Submitted by Trupti Ranjan Behera 710CH1020 Under the Supervision of Dr. Basudeb Munshi In partial fulfillment for the award of the degree of Masters of Technology (Dual Degree) In Chemical Engineering Department of Chemical Engineering National Institute of Technology Rourkela May 2015

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Thesis

On

A Computational Study on Entropy Generation

Minimization of Pipe Bending

Submitted by

Trupti Ranjan Behera

710CH1020

Under the Supervision of

Dr. Basudeb Munshi

In partial fulfillment for the award of the degree of

Masters of Technology (Dual Degree)

In Chemical Engineering

Department of Chemical Engineering

National Institute of Technology Rourkela

May 2015

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NATIONAL INSTITUTE OF TECHNOLOGY

ROURKELA

CERTIFICATE

This is to certify that the project entitled “A Computational Study on Entropy Generation

Minimization of Pipe Bending” submitted by Trupti Ranjan Behera (710CH1020) in partial

fulfillment of the requirements for the award of Master of Technology (Dual degree) in Chemical

engineering, Department of Chemical Engineering at National Institute of Technology, Rourkela

is an authentic work carried out by him under my supervision and guidance.

To the best of my knowledge the matter embodied in this thesis has not been submitted to any

other university/Institute for the award of any Degree.

Dr. Basudeb Munshi

Date: Department of Chemical Engineering

Place: Rourkela NIT Rourkela

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ACKNOWLEDGEMENT

On the submission of my thesis entitled “A Computational Study on Entropy Generation

Minimization of Pipe Bending” I would like to express my deep and sincere gratitude to my

supervisor Dr. Basudeb Munshi, Department of Chemical Engineering, for his guidance constant

motivation, valuable suggestions, encouragement and support during the course work.

I would like to take the opportunity to thank Mr. Akhilesh Khapre for his unconditional

assistance and support. I would like to thank again Dr. Basudeb Munshi for arranging

computational facility for numerical simulations.

I would offer my loving thanks to my friends Aslam, Ramya, Devipriya and Pallav who provided

me with strength and moral support. I would like to thank all others who have consistently

encouraged and gave me moral support, without whose help it would be difficult to finish this

project.

Last but not the least, I would like to express my love, respect and thanks to my parents Mrs

Kanchan Late Giri and Mr. Kshiroda Kumar Behera for being backbone of my life and for their

unconditional love. Special thanks to my sister and brother.

Trupti Ranjan Behera

710CH1020

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ABSTRACT

This project presents solution of a simple optimization problem of flow through circular bent

channels. Six different bending angles of the pipe were taken for finding the effect of bending

angle on the pressure drop. The diameter and length of all the pipes were taken same irrespective

of the bending angles. The CFD model equations were solved to predict the hydrodynamic

behaviour of pipe. The models were solved by ANSYS Fluent 15.0 solver. The CFD models

were validated by comparing the computed pressure drop with the analytically computed

pressure as given in the previous literature. The entropy generation rate was compared and the

optimum bending angle was found by taking pressure drop across the pipe as reference. The

entropy generations in both isothermal and non-isothermal conditions were studied and

compared mutually for a particular flow rate. An EGM analysis was carried out and a

thermodynamically optimal solution was identified. It was also observed that as the bending

angle increases the entropy generation decreases for a particular flow rate.

Keywords: CFD, Entropy generation minimization, Pipe bending

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Contents

ABSTRACT ................................................................................................................................... iii

LIST OF FIGURES ....................................................................................................................... vi

LIST OF TABLES ....................................................................................................................... viii

NOMENCLATURE .................................................................................................................... viii

CHAPTER-1 INTRODUCTION .................................................................................................... 1

1.1 Energy .............................................................................................................................. 1

1.2 Entropy ............................................................................................................................. 1

1.3 Entropy Generation Minimization (EGM) ....................................................................... 1

1.4 Mass transfer Studies using EGM .................................................................................... 2

1.5 Nano-Fluid Flow and heat transfer Using EGM .............................................................. 2

1.6 Heat transfer and flow through a channel using EGM ..................................................... 2

1.7 Motivation ........................................................................................................................ 3

1.8 Objective of the work ....................................................................................................... 3

CHAPTER-2 LITERATURE REVIEW ......................................................................................... 5

2.1 Pressure drop in Channels ................................................................................................ 5

2.2 Entropy Generation analysis in Mass Trasfer Applications ............................................. 5

2.3 Entropy Generation analysis of Nano-Fluid Flow ........................................................... 5

2.4 Flow through Microchannels ........................................................................................... 6

2.5 EGM analysis for pipe or channel flow ........................................................................... 8

2.6 Entropy Generation analysis for Heat exchangers ......................................................... 12

CHAPTER-3 GOVERNING EQUATIONS ................................................................................ 14

3.1 COMPUTATIONAL FLUID DYNAMICS: ................................................................. 14

3.2 Laminar and turbulent flow into the pipe: ...................................................................... 16

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3.3 Governing equations for entropy generation:................................................................. 17

3.4 Equations for Pressure drop ........................................................................................... 18

CHAPTER 4 Specification of Problem ........................................................................................ 20

4.1 Problem specification and Geometry Details: ................................................................ 20

4.2 Meshing of geometry: .................................................................................................... 21

4.3 Physical Model ............................................................................................................... 22

4.4 Material Properties ......................................................................................................... 22

4.5 Boundary Conditions: .................................................................................................... 22

4.6 Method of solution: ........................................................................................................ 23

Chapter-5 Results and Discussion ................................................................................................ 24

5.1 Validation ....................................................................................................................... 24

5.2 prediction of entropy generation in pipefor isothermal boundary condition ................. 26

5.3 Non-isothermal boundary condition.................................................................................... 31

5.3.1 Prediction of Entropy generation for Constant heat flux wall ............................... 31

5.3.2 Prediction of Entropy generation for Constant wall Temperature ......................... 39

Chapter-6 Conclusion ................................................................................................................... 46

References: .................................................................................................................................... 47

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LIST OF FIGURES

Figure 1 Geometry of the Pipe ...................................................................................................... 20

Figure 2 Geometry for α =600 ....................................................................................................... 21

Figure 3 Meshing for α= 600

bend pipe ........................................................................................ 21

Figure 4 Pressure drop versus Reynolds number plot for α =450 ................................................. 24

Figure 5 Pressure drop versus Reynolds number plot for α =900 ................................................. 25

Figure 6 Pressure drop versus Reynolds number plot for α =180 0 .............................................. 26

Figure 7 Contours of static pressure of α=900 at point b of the channel for (I) Reynolds no 1000,

(II) Reynolds no 5000 (III) for Reynolds no 20000 ...................................................................... 27

Figure 8 Velocity contours for α = 900 at point b of the channel for (I) for Reynolds number

1000, (II) for Reynolds number 5000, (III) for Reynolds no 20000 ............................................. 28

Figure 9 Contours of local entropy generation values for α = 900at different points of the channel

for (a) Reynolds no 1000 (b) Reynolds no 5000 (c) Reynolds number 20000 ............................. 29

Figure 10 Curve between Reynolds number and volumetric entropy generation for different α . 30

Figure 11 Temperature contour α = 1200 and wall flux = 10000W/m

2 at different points of the

channel for (I) Re= 1000, (II) Re=12000, (III) Re=30000 ........................................................... 32

Figure 12 Local thermal dissipation rate for bending angle 1200and wall flux 10000W/m

2at

different points of the channel for(I) Re= 1000, (II) Re=12000 ................................................... 33

Figure 13 Contours of Local viscous dissipation rate for α = 1200and wall flux 10000W/m

2 at

different points of the channel for (I) Re= 1000, (II) Re=12000, (III) Re=30000 ........................ 34

Figure 14 Contours of Entropy Generation for α = 1200 and wall flux 10000W/m

2 at different

points of the channel for(I) Re= 1000, (II) Re=12000, (III) Re=30000 ....................................... 35

Figure 15 Curve between Reynolds number and volumetric entropy generation for α = 1200

angle of pipe bending at wall flux 10000W/m2. ........................................................................... 36

Figure 16 Curve between Reynolds number and volumetric entropy generation for different α at

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wall flux 1000W/m2 ...................................................................................................................... 36

Figure 17 Curve between Reynolds number and volumetric entropy generation for different α at

wall flux 5000W/m2 ...................................................................................................................... 37

Figure 18 Curve between Reynolds number and volumetric entropy generation for different α at

wall flux 10000W/m2 .................................................................................................................... 37

Figure 19 Curve between Reynolds number and volumetric entropy generation for different α at

wall flux 20000W/m2 .................................................................................................................... 38

Figure 20 Entropy Generation versus Re plot for different wall flux for α =1200 ....................... 39

Figure 22 Contours of Viscous dissipation for 600 bend wall temp=315K at different point-b of

the channel (I) Re= 1000, (II) Re=11000, (III) Re=20000 ........................................................... 40

Figure 23 Contours of Entropy Generation for α = 600 and wall temperature=315Kat different

points of the channel for(I) Re= 1000, (III) Re=20000 ................................................................ 41

Figure 24 Curve between Reynolds number and viscous dissipation and thermal dissipation for α

= 600 at wall Temp 315K .............................................................................................................. 42

Figure 25 Curve between Reynolds number and volumetric entropy generation for different α at

wall temp 305K ............................................................................................................................. 43

Figure 26 Curve between Reynolds number and volumetric entropy generation for different α at

wall temp 310K ............................................................................................................................. 43

Figure 27 Curve between Reynolds number and volumetric entropy generation for different α at

wall temp 315K ............................................................................................................................. 44

Figure 28 Curve between Entropy Generation and Re plot for different wall temperature for α

=600 ............................................................................................................................................... 44

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LIST OF TABLES

Table 1: Constant values for 3-K method ..................................................................................... 19

Table 2: Choice of model based on Reynolds number ................................................................. 22

Table 3: Property of fluid water .................................................................................................... 22

Table 4: Check For Mesh Independency by RMS error for straight pipe (α=1800) ..................... 25

NOMENCLATURE

S= Entropy

CFD= Computational Fluid Dynamics

Re= Reynolds Number

Α= Bending Angle

EGM= Entropy Generation Minimization

= Volumetric Entropy Generation

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CHAPTER-1

INTRODUCTION

1.1 Energy

Energy is everywhere. Anything we eat or use has energy in it. Everything that we process or

every object we produce requires energy, and the economic growth of a country is related to the

energy demands. But during an engineering operation most of the energy is not utilized so it

transforms to another form and loss of energy occurs. As the world energy situation is getting

more and more serious, the optimization of thermal systems attracts increasing attention because

about 80% of the total energy consumption is related to heat and improving the performance of

thermal systems is of great significance to the energy conservation [1].

1.2 Entropy

Entropy is the degree to which energy is wasted. While the exergy of a system is the maximum

work possible during a process. So the amount of available work (or exergy) depends on the

amount of entropy produced [2]. The amount of entropy generated can be directly used as an

efficiency parameter of the system. Minimization of entropy generation in a thermodynamic

system provides efficient use of exergy that is available.

1.3 Entropy Generation Minimization (EGM)

The theorem of minimum entropy generation says that, under certain assumptions, the global

entropy production rate of a given system attains a minimum value when the processes in the

system become stationary. This method combines the most fundamental principles of

thermodynamics, heat transfer, and fluid mechanics [2].

Bejan [2] studied minimization of entropy generation in heat exchangers, insulation systems,

storage system, power generation, refrigeration processes in brief. But later on many more

detailed analysis of his work is carried out by many authors.

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1.4 Mass transfer Studies using EGM

Like in mass transfer operations, Barbosa et al. [16] have optimized the finned tube evaporator

using the Entropy generation minimization (EGM) principle. While Özyurt et al. optimized

evaporator for use in a refrigeration cycle [17]. For condensation of gases in the presence of high

non-condensable gases, in the design of fan-supplied tube-fin condensers and Refrigerant

circuitry design of tube condenser different authors used EGM as efficiency parameter [18-20].

Ng et al., [21] analyzed the EGM of absorption chiller. Mistry et al. [22] used entropy generation

as a parameter to analyze the desalination process.

1.5 Nano-Fluid Flow and heat transfer Using EGM

In nanofluid flow, Leong et al. [41] and [42] analyzed entropy generation of three different types

of heat exchangers studying the turbulent convection of nanofluids subjected to constant wall

temperature. Singh et al. [44] And Bianco et al. [43] theoretically investigated entropy generation

of nanofluids convection and showed the existence of different optimal working points according

to the flow features without considering the influence of particles diameter. Bianco et al. [45]

numerically developed turbulent forced convection flow in a square tube, subjected to constant

and uniform wall heat flux of a water–Al2O3 nanofluid. Bianco and Manca [46] proposed a

simple analytical procedure to evaluate the entropy generation and showed that it takes a good

agreement with the numerical calculations.

1.6 Heat transfer and flow through a channel using EGM

Sahin [5] considered a range of laminar streams at steady heat flux boundary condition and

explored different pipe geometries to minimize the entropy losses. Among different geometries,

when the frictional contribution of entropy generation are prevailing round geometry is the best.

Dagtekin et al.[6], Oztop [7] studied the effect of longitudinal blades of diverse shapes for

laminar stream in a circular conduit and demonstrated that the cross-sectional region and the wall

heat flux have extensive impact on entropy generation. For improved heat exchange surfaces of a

tubular heat exchanger at steady wall temperature Zimparov [8] concentrated on liquid

temperature variety along the length of exchanger by creating execution assessment criteria

mathematical equations taking into account the entropy generation hypothesis. Mahmud and

Fraser [9] concentrated on channel with circular cross-sectional area and between two parallel

plates and investigated second law attributes of heat exchange and liquid stream because of

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constrained convection. Sahin [10] demonstrated that consistency variety had an extensive

impact on both entropy generation and pumping power. Thermodynamic enhancement of

convective heat move in a funnel with consistent wall temperature has been considered by

Mukherjee [11], Esfahani and Alizadeh [13] and Sahin [12]. Esfahani They discovered a

relationship between ideal estimations of Stanton number and the proportion of the length of the

funnel to its width (St.L/D). Sahin et al. [14] numerical dissected the impact of fouling on

operational cost in channel stream on entropy generation. Sahin [15] additionally upgraded the

measure of heat exchanger and the channel temperature of the liquid for which the aggregate

irreversibility because of heat exchange and weight drop turns into the base by examining

entropy generation for a laminar thick stream in a conduit subjected to consistent divider

temperature. Esfahani et al. [4] broke down the impact of non-uniform heating in laminar stream

on entropy generation.

1.7 Motivation

. There is an increased need for utilization of a variety of duct geometries for internal flow heat

transfer applications along with the advancement of science and technology. Due to the size and

volume constraints in the areas of electronics, biomedical, atomic, and aviation, it may be

required to use different flow passage geometries.

Up to now several studies have been done taking geometrical factors into account to calculate the

Entropy generation rate. For example Sahin [5] took different geometries like rectangular,

circular, triangular, sinusoidal and square to analyze the flow through the channel. Hilbert and

Mercus [36] [37], calculated the pressure drop in tee bending, short radius elbow and long radius

elbow of channel flow but considered only bending angle to be 900. Sciubba [35] studied the

viscous dissipation effect of bifurcation of a channel on entropy generation.

But no one has studied the effect of bending angle on pressure drop and entropy generation. So

this study is to analyze the effects of various bending angles on different parameters.

1.8 Objective of the work

Validation for pressure drop across the pipe inlet and outlet for various Re number.

To find out the effect of Re on pressure drop across the pipe ends

To find out the effect of Re on entropy generation for iso-thermal (constant temperature

and same of liquids and wall) problems.

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To find out the effect of Re on entropy generation for constant heat flux boundary

condition case.

To find out the effect of Re on entropy generation for non-isothermal constant wall

temperature case.

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CHAPTER-2

LITERATURE REVIEW

2.1 Pressure drop in Channels

Hilbert[36]

examined three bends: blinded tee, short radius elbow and long radius elbow

experimentally. He found that from the standing point of the wear the blinded tee is the best bend

followed by short radius elbow and long radius sweep is the least preferable one.

Marcus et al.[37]

measured the pressure loss of the three bends and found that for a wide range of

mass flow ratio short radius elbows provides the lowest pressure loss for fine powdered

materials.

2.2 Entropy Generation analysis in Mass Trasfer Applications

Lienhard et al. [22] analyzed desalination process considering the separation work along with it

lost work due to entropy generation. Entropy generated due to irreversibility in the separation

process, along with the irreversible mixing of the brine with the ambient seawater and

temperature disequilibrium of the discharge. The entropy generation due to chemical

disequilibrium was found to be important for systems with high recovery ratios.

2.3 Entropy Generation analysis of Nano-Fluid Flow

Manca et al. [45] investigated the entropy generation in turbulent forced convection flow of

Al2O3–water nano fluid at constant wall heat flux boundary condition. For a constant Reynolds

number, nanoparticles were added to decrease entropy generation. The optimal concentration

reduced with increase of Re. The reduction of velocity decreased entropy generation due to

friction. Total entropy generation was reduced by very small concentration of the nano-fluids due

to the increase of nano-fluid thermal conductivity. When constant inlet velocity condition was

considered, by increasing the concentration of the nano-fluid, entropy generation rate was

increased, because viscosity effect began to predominate, Re was reduced and consequently total

entropy generation rate was increased.

Singh et al. [44] analysed entropy generation of nanofluids of two different viscosity models,

while considering conductivity, for laminar and turbulent flow. They have found that an

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optimum diameter at which the entropy generation rate was occurred was less for laminar flow in

comparison to turbulent flow. The entropy generation ratio was found to be always greater than

one and it increased with volume fraction for laminar flow. But the behaviour was found to be

opposite for turbulent flow. The viscous entropy generation part decreased with volume fraction,

while thermal part increased with volume fraction. At smaller diameter, the viscous entropy

generation part has higher values and at higher diameters, the thermal part of entropy generation

has. The entropy generation rate increased very slowly or became constant after a particular

diameter.

Leong et al. [41] analyzed the entropy generation of nanofluid. As the volume fraction of the

nanaoparticle was increased total dimensionless entropy generation was decreased. When

Titanium dioxide nanofluids and alumina nano-fluids were compared for lower dimensionless

entropy generation value, titanium dioxide nanofluids showed good results. As the

dimensionless temperature difference was increased, total dimensionless entropy generation also

increased. Mass flow rate of Titanium dioxide or alumina nanofluid influenced the total

dimensionless entropy generation.

Leong et al. [42] also studied different types of heat exchangers using nanofluids as working

fluid. Shell and tube heat exchanger with case of helical baffles (25°) showed highest overall

heat transfer coefficient. But when the mass flow rate was increased the result also came the

same followed by segmental and helical baffles (50°). When the nanoparticles were added the

overall heat transfer coefficient was increased but it was found to be independent of the types of

heat exchanger. The HTC for heat exchanger with 25° helical baffles was the highest so the heat

transfer rate compared to that of segmental and 50° helical exchanger. Entropy generation was

minimized for the 50° helical exchanger due to its lowest total pressure drop.

2.4 Flow through Microchannels

Ibáñeze et al. [30] optimized various flows in a parallel plate microchannel having finite wall

thickness using the entropy generation minimization method. An optimum slip velocity was

obtained which was leading to a minimum entropy generation rate and they analyzed effects of

slip velocity on the optimum values of some other parameters. Besides all these the Nusselt

number was also calculated and analyzed. An optimum value of the slip length for which the

heat transfer was maximized was derived.

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Ibáñez et al. [31] illustrated the combined effects of slip flow and wall heat on entropy

generation in a microchannel. The entropy generation reached a minimum for the wall heat flux

for geometrical and physical parameters of the system and wall-fluid thermal conductivity ratio.

The optimum values of both the slip flow and wall to fluid thermal conductivity ratio, where the

entropy generation was minimum, decreased with the wall heat flux. Finally, these minimum

values of entropy, reached in the optimum values of slip flow and wall to fluid thermal

conductivity ratio.

Hung et al. [32] developed analytical model of water-alumina nanofluid flow in circular

microchannels and investigated the effects of nanoparticle suspension, microchannel aspect ratio,

and stream-wise conduction in low-Peclet-number flow regime on the entropy generation.

Stream-wise conduction affected the thermal characteristics of the nanofluids, and consequently

the entropy generation but its effect due to fluid friction irreversibility was not significant. When

nanoparticle suspension increased the heat transfer irreversibility was dominant. In certain range

of low Peclet number flow at optimum condition for nanofluid flow in microchannels minimal

entropy generation was identified. The aspect ratio of the microchannel affected the second-law

performance of nanofluid prominently. By increase in the aspect ratio was found to effectively

enhance the exergetic effectiveness of the flow when the heat transfer irreversibility was

dominant. When the fluid friction irreversibility was dominant, the choice of working fluid was

more important than the design of the microchannel.

Anand [33] analyzed the thermally fully developed flow with viscous dissipation of a non-

Newtonian fluid through a microchannel. The effect of slip on heat transfer and entropy

generation was important and could not be neglected, especially for shear thickening fluids

(n > 1) the curves for velocity distribution and temperature distribution had higher slopes,

especially towards the centerline of the channel, in comparison to shear thinning fluids. The slip

parameters only affected the advection of fluid momentum and not its diffusion For symmetrical

slip boundary conditions. Average entropy generation rate was lower than the corresponding

values predicted by asymptotic slip law for shear thickening fluids; and this difference increased

with increase in the value of slip coefficients. The Bejan number for shear thickening fluids was

much less than that for shear thinning fluids and Be < 0.5 throughout the channel width so we

can say that for shear thickening fluids, most of the irreversibility was due to fluid friction. Thus

to minimize the entropy generation for power law fluids, for shear thickening fluids

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concentration should be on friction irreversibility and for shear thickening fluids on heat transfer

irreversibility. For shear thinning fluids (n<1), Bejan number tends to 1 at axis of symmetry,

while for shear thickening fluids (n>1), Bejan number tends to 0.

Huai et al. [34] numerically investigated the effect of viscous dissipation on total entropy

generation for curved square microchannles in the laminar flow region by using two different

working fluids. The working fluids were aniline and ethylene glycol. The effect of viscous

dissipation was studied and analyzed to calculate the total entropy generation number and heat

transfer entropy generation number. When the case of cooled fluid was considered heat transfer

entropy generation number and the frictional entropy generation number were decreased due to

the viscous dissipation effect. The Bejan number was found to be lower when the working fluid

aniline was heated than to cooling. The curved microchannels having smaller curvature radius

showed earlier total entropy generation number minimization, and the Bejan number generally

reduced as the curvature radius diminished. The heat transfer entropy generation number

decreased with Reynolds number. The total entropy generation number extremum did not appear,

for the case of ethylene glycol at large Reynolds number region when it was heated under

viscous dissipation effect. Throughout the study they have found that the frictional entropy

generation dominated the total entropy generation.

2.5 EGM analysis for pipe or channel flow

A number of studies have been done on entropy generation in fluid flow and heat transfer

problems. Bejan showed that the entropy generation is due to heat transfer and viscous friction

for forced convection viscous fluid flow in a channel. He also showed that when the entropy

generation is minimized, for the case of heat transfer in a circular tube, a relation exists between

heat transfer and viscosity effects of the components of entropy generation associated with it.

Bejan also applied the concept of minimum entropy generation to the design of a counter-flow

heat exchanger. He showed that an optimum flow path length would be possible [3] [4] [2].

Bertola and Cafaro [23] reviewed the principle of minimum entropy production of fluid at rest by

the heat conduction and the combined effect of heat conduction and shear flow in an

incompressible fluid flow. This theorem was a best approximation method and as the system

converged to equilibrium it converged to the exact solution. And for a system in a stationary

state the entropy production was approximately zero.

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Sahin [5] compared entropy generation and pumping power required for different duct geometry

for optimum shape with constant wall temperature boundary condition. Duct geometries used

are: square, circular, rectangle, equilateral triangle, and sinusoidal. The optimum duct geometry

depends on the Reynolds number. Circular duct geometry proves to be the optimum when the

entropy generation was mostly contributed by frictional losses. Rectangular duct would gave the

best result when the ratio between length to width or the aspect ratio became one. From the point

of view of entropy generation for low Re number region sinusoidal duct geometry appeared to be

the best, but the pumping power requirement for the sinusoidal geometry was higher than that for

the square geometry.

Vučković et al. [24], studies on the numerical simulation of local entropy generation rate in flow

meter and pipeline curve. The results of numerical simulation show good agreement with

measured data in regard to temperature. The friction factor of irreversibility is more influence on

the generation of entropy than heat transfer irreversibility.

Mahmud and Fraser [9] studied the forced convection inside channel with circular cross-section

and channel of parallel plates having finite gap in between them at iso-flux and isothermal

boundary conditions. Along with the Newtonian fluid they considered Non-Newtonian fluid flow

in the above boundary condition. Entropy generation number and Bejan number were derived

analytically for each case of axial conduction, heat transfer normal to the axis and fluid frictional

irreversibility. And then he studied the Bejan no/entropy gen in axial, radial and different

directions.

Sahin [10] studied numerically the entropy generation rate in a developing laminar viscous fluid

flow in a circular pipe with variable viscosity and constant heat flux boundary condition. The

entropy generation rate is higher near the wall and decreases along the radius away from the

surface of the pipe as the temperature and viscosity gradient are high near the wall of the pipe.

Around the centerline of the pipe viscosity and temperature gradients are small and hence the

entropy generation was small. The viscosity variation affects velocity and the temperature

profiles, as well as the entropy generation rate. Axial variation of the entropy generation rate

near the wall sharply increased to a maximum and then steadily decreased. The integrated value

of entropy generation rate increased over the cross-sectional area of the pipe due to the

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temperature penetration and the widening of the thermal as well as hydrodynamic boundary

layers.

Abolfazli and Alizadeh [13] have also studied the thermodynamic optimization of geometry in

convective heat transfer at constant temperature at the wall by considering laminar flow. They

found out the effects of different parameter on the pumping power and Entropy generation. They

also introduced a proper correlation for the optimum design of the tube.

Sahin [12] wrote the dimensionless entropy generation and pumping power to heat transfer ratio

as functions of three dimensionless numbers, using an average heat transfer coefficient and a

friction factor at average bulk temperature. The entropy generation per unit heat transfer rate

decreased with duct length, while the entropy generation per unit capacity rate increased. On the

other hand when the variation of viscosity with temperature was considered, the pumping power

to heat transfer ratio decreased with duct length. As the dimensionless wall heat flux is increased,

entropy generation per unit heat transfer rate and entropy generation per unit heat capacity

decrease initially reaching a minimum for the case of variable viscosity model and then increase.

As the wall heat flux decreases, pumping power to heat transfer ratio may increase very sharply

for viscous fluids.

Sahin [15] studied entropy generation rate in a developing laminar viscous fluid flow in a

circular pipe with iso-flux boundary condition. Near the wall the entropy generation rate was

higher and sharply decreases along the radius away from the surface of the pipe. This happens so

because of the existence of the velocity and temperature gradients in the region of the pipe.

Around the centerline entropy generation rate was small because of the small temperature and

velocity gradients. Thermal component contribution to the entropy generation rate was high.

Along the axial direction near the wall the entropy generation rate was very high and reaches

maximum then decreases steadily. Due to flux, temperature penetration occurred into the

working fluid. Along with it hydrodynamic boundary layer and thermal boundary layer got

widened. So the entropy generation rate along the axial direction over the cross sectional area of

the pipe increased steadily.

Sahin [14] also studied the effect of fouling due to heat transfer and pressure drop in viscous pipe

flow on the irreversibility components. He found that the increase in the irreversibility due to

heat transfer was slower than that due to viscous friction. As the fouling layer build up, total cost

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due to irreversibility reached a minimum and after that it increased. The fouling layer thickness

depends upon the cost ratio. The optimum thickness dropped as the cost ratio was increased. The

cost of irreversibility was found to be dependent and directly proportional to the temperature

difference between the tube wall and inlet fluid. He suggested that by considering the cost

variation versus fouling layer thickness excessive cost due to the irreversibility can be avoided.

Jankowski [27] analyzed and generalized entropy generation in fully-developed convective heat

transfer flow. He made the correlations to determine the optimum cross-sectional shape of a flow

passage. The circular cross-section minimized flow resistance hence the entropy generation

minimization occurs. From the heat transfer correlations he suggested that the circular cross-

section was not always ideal. For a duct with a large wetted perimeter has more surface area

available for heat transfer so the heat transfer irreversibility component dominated and

minimization of the overall entropy generation occurred.

Sahin and Yilbas [28] studied the flow between the parallel plates and of the bi-vertical motion

of the top plate and its effect on entropy generation. By increasing the velocity of the plate, the

entropy generation rate lowers. When they kept the velocity constant the entropy was found to

take shape of an exponent and when the force was kept constant the entropy took the shape of a

parabola. In the flow system combination of those two steps results in the minimum entropy

generation. At specific displacement value minimum entropy occurred. Constant velocity

operation should was allowed to reach a particular gab height of the plates before the constant

applied force. Consequently, combination of the two cases provided the minimum power loss

and minimum entropy generation in the working fluid between the two plates considered.

Sciubba.[35] studied the entropy generation rate due to viscous dissipation for three cases: (a) a

single bifurcation, (b) a straight tube with wall suction, and (c) bifurcation with wall suction. For

laminar bifurcation without losses the entropy generation rate depended strongly upon the aspect

ratio. For any bifurcation length, constant velocity case showed a lower entropy generation rate

than the constant Re case. For each physical situation of constant Re and constant inlet velocity a

optimal configuration was existed and it displayed minimum value of the entropy generation

rate. For the examined situations, a configuration existed that showed the lowest viscous entropy

generation rate which was found to be compatible with the imposed constraints.

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Jarungthammachote [29] studied, the entropy generation of transient heat conduction with fixed

temperature boundary conditions. For heating case the maximum non-dimensional entropy

generation rate takes place near the surface and its value reduces with increase of time because of

high temperature gradient near the surface. Initial temperature has very weak effect on total

entropy generation rate. When time increases the effect of initial condition disappears and as the

time approaches infinity, the temperature distribution, the local entropy generation rate, and the

total entropy generation rate of transient problem are approach those of steady state problem.

2.6 Entropy Generation analysis for Heat exchangers

Dagtekin et al.,[6] Measured the entropy generation and the pumping power variations for

laminar flow and constant wall temperature conditions in a circular duct with longitudinal thin,

triangular and V-shaped fins respectively. The inlet to wall temperature difference was found to

be a function of entropy generation, and increased with increase if the temperature difference. As

the number of the fins or length of the fins was increased, the dimensionless entropy generation

was also increased. When the higher values of the Reynolds number were considered, the

entropy generation for thin fins was found to be superior to triangular fins. While triangular fins

were considered by gradually increasing the fin angle, entropy generation to heat transfer ratio

increased. While for V-shaped fins, with the increase of the fin angle the entropy generation and

the pumping power to heat transfer ratio both were increased.

Tandiroglu [25] investigated a transient turbulent flow in a circular pipe with baffle inserted into

it and studied the characteristics of forced convection and entropy generation with constant heat

flux condition and proposed a correlation. He suggested that the baffle inserted tubes of type

4562 to be preferred for the best performance because it exhibited the minimum time averaged

entropy generation number for transient forced convection flow conditions based on

experimental data and correlations for time averaged entropy generation number.

Oztop [7] analyzed the second law of thermodynamics of laminar flow subjected to constant wall

heat flux for semi-circular ducts. As Reynolds number or temperature difference is increased

total entropy generation is decreased. Contingent upon Reynolds number and when cross-

sectional area is expanded aggregate entropy generation is expanding for the same Reynolds

number. Further, an increment in cross-sectional area reasons to build obliged pumping power.

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Zimparov [8] studied heat transfer and fluid friction characteristics of 10 spirally corrugated

tubes to illustrate the application of the extended performance evaluation criteria equations. The

results for different design constraints show that the optimum rib-height-to-diameter ratio (e/D)

for spirally corrugated tubes is about 0.04 and the entropy generation value is minimum for this

case. An augmentation technique is utilized in a heat exchanger to improve its performance

characteristics, more significant benefits can be obtained if it operates at constant wall

temperature.

Nag and Mukherjee [11] studied heat transfer from a duct with constant wall temperature during

phase change on one side of a heat exchanger tube. The important design criteria was the initial

temperature difference The heat transfer to pumping power ratio was used at its optimum value.

Simply maximizing this ratio of pumping power to heat transfer is not often a good solution,

because sometimes a large amount of energy may be irreversibly lost and the entropy generated

may be far from the minimum possible.

The common closed thermodynamic systems like Carnot cycle and refrigeration cycles etc. are

analyzed with the concepts of dimensionless entropy generations by Cheng and Linag [26]. They

found that when the entropy generation number was small the larger is the exergy-work

conversion efficiency. It is found that a percentage of the given mechanical work turns to be the

net exergy stream into the heat sources, while the rest is destructed in view of entropy

generation. At the point when the net exergy stream rate into the heat sources is altered, littler

entropy generation rate leads littler information force, while it prompts bigger net exergy stream

rate into the heat energy for settled input power.

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CHAPTER-3

GOVERNING EQUATIONS

3.1 COMPUTATIONAL FLUID DYNAMICS:

Having the capacity to concoct proficient and ideal outlines for new items is one of the greatest

difficulties in the building processes. One of the most grounded devices in this classification is

FLUENT. Familiar is an exceptionally helpful system procured by ANSYS. It has the ability to

model liquid move through articles, past items and numerous more and the capacity to test, break

down results and outline under one system.

ANSYS Familiar is Computational Liquid Motion (CFD) programming. It permits users to

simulate flow issues of running multifaceted nature. It has wide physical modeling abilities by

which planning of the model flow, turbulence, heat transfer, and reactions over objects is done

by the user. From the utilization of CFD programming as a fundamental piece of their outline

stages a great many organizations has been profited in their item advancement. It utilizes the

finite volume method to comprehend the representing Navier-Stokes mathematical statements for

a liquid. Navier-stokes equations are gotten from (1) the mass conservation equation, (2) the

preservation of energy equation and (3) the preservation of momentum equation.

( ) 1

( ) 2

∫ ∫

3

The preservation of mass, momentum and energy are coupled and non-linear arrangement of

differential mathematical statements. It is for all intents and purposes difficult to tackle the above

comparisons systematically for useful engineering issues. So the difficulty arises. But CFD

software FLUENT provides approximation to solve the specified governing equations.

FLUENT also allows to model flows such as compressible or incompressible, viscous or invisid,

turbulent or laminar flow. It has flexible moving and deforming meshes through which fast and

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accurate results are obtained to create optimal designs. Ultimately, FLUENT allows engineers to

design, create and analyze a configuration all under one program.

To model the object that a user wants to work with, its geometry and mesh must be first created

in ANSYS Workbench and the meshing implementation can be done in the specimen to analyze

the structure as they try to analyze fluid flow through or past the object.

A couple of diverse methods for displaying and investigating liquid flow are through turbulence

modeling, k-, and Y+. To model turbulent flow turbulence modeling is used. Large, nearly

random fluctuations in velocity and pressure in both space and time are the characters of

turbulent flow. The fluctuations arised from the turbulent instability are dissipated into heat by

viscosity or viscous dissipation. In the opposite limit of high Reynolds numbers turbulent flows

occur. There are two approaches to solving the turbulent flow equations, direct numerical

simulations and k-. In immediate numerical simulation the Navier-Stokes equations is

specifically integrated, determining the majority of the spatial and transient changes without

turning to displaying yet in k-, models Reynolds push in two turbulent parameters, the turbulent

statements 4 and 5 respectively.

( ) 4

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

5

As the complexity in the geometries of the design increase it is required to adjust the meshing

accordingly. The CFD software codes solve directly the flow variables at each element and the

values at other locations are interpolated. Where there more flow gradients occurring around a

certain area rather than a complicated geometry one can concentrate the mesh around a certain

area.

Enhanced wall treatment is another modeling capability FLUENT uses. It is especially used for

turbulent cases using the k-epsilon model as it analyzes the object closer near the wall region.

The initial and boundary conditions are specified in FLUENT. And after initializing the problem;

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convergence is checked. If the convergence is not achieved accurate results will not be obtained.

Finally, FLUENT provides a wide variety of parameters that can be plotted and analyzed.

Let us consider the finite-difference method in 1D. If the grid has equally-spaced points with

being the spacing between successive points, the truncation error is O ( ). As the number of

grid points is increased, and the spacing between successive points is reduced, the error in the

numerical solution would decrease. Therefore the obtained numerical solution will closely agree

to the exact solution. In FLUENT for the convergence the governing equations are solved for a

number of iterations. The magnitude of the average of variable is computed by Navier Stokes

equation and Equation 6.

√∑

6

where u indicates a particulate variable to be computed, N is the number of iterations to be

performed, R is the residual, and the subscript g indicates a guessed value.

3.2 Laminar and turbulent flow into the pipe:

FLUENT is a very powerful tool in obtaining solutions to a wide range of fluid flow problems.

The results obtained should be carefully validated with known theory or empirical data to make

sure they are accurate. For example, for the flow through a pipe, the pressure drop obtained can

be compared with the empirical formulae. For a steady state laminar flow and turbulent flow in

circular tubes the Navier-Stokes upon making the necessary assumptions can be solved. The

incompressible Navier-Stokes equations in Cartesian coordinates are shown in Equations 7, 8

and 9.

(

)

(

) 7

(

)

(

) 8

(

)

(

) 9

The parabolic velocity profile for steady laminar flow in a cylindrical pipe is:

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10

11

Depending upon the accurateness of experimental results and empirical correlations, whether to

improvement meshing or whether or initial conditions or the boundary ones is decided. The

reason of high dependency on empirical data is due to the association of randomness with

turbulence. In turbulence there is no exact equation, all solutions are empirical data points. An

example of this is the equation for skin friction for turbulence which is

12

The main reason that it is an empirical equation, it has a few limiting conditions. Some of those

limiting conditions are that it has to be only for smooth pipes and it is only useful for Reynolds

number less than 100,000.

Also, another very important one is Nikuradse‟s empirical correlation for turbulent flow which is

given by

18

where n, the power-law exponent varies with respect to the Reynolds number. Turbulent cases

are very difficult to analyze because there is no exact answer, however the Moody diagram in

which the coefficient of friction with respect to the Reynolds number is displayed, serves as a

reference as to what the solutions should appear to be. The Moody diagram is based off of

19

where the results are obtained from numerous set of experiments plotted on the Moody diagram.

3.3 Governing equations for entropy generation:

The equations used in CFD modeling are continuity equation, momentum equation and energy

equation, which are all previously mentioned.

Equation used to calculate the entropy generation by viscous dissipation using a custom field

function in CFD modeling is

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(

) (

) 20

Where

{(

)

(

)

(

)

} (

)

(

)

(

)

21

And

{(

)

(

)

(

)

} 22

The entropy Generation term consists of two parts: Viscous dissipation and Thermal

Dissipation.And again the viscous dissipation term consists of two parts, one is normal viscous

dissipation (

) and another one is turbulent dissipation rate (

. Turbulent dissipation rate is

due to the turbulence in the flow.

(

) 23

In isothermal case only viscous dissipation term comes into action. And for non-isothermal case

along with the viscous term, thermal term is there.

(

). 24

3.4 Equations for Pressure drop

Empirical method is used to calculate the pressure drop using the following equations

For straight pipe the Pressure drop is calculated as:

25

Or Hagen- Poiseuille equation for laminar flow

26

where f is the friction factor

for laminar flow

27

for turbulent flow

√ (

√ ) 28

For bending pipes head loss is calculated using the 3-K method and the pressure drop is

calculated

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Darby’s 3-K method

(

) 29

Table 1: Constant values for 3-K method

angle in0.3 mm

0.3

900, mitered one weld 1000 0.27 4 10.6

450, mitered one weld 500 0.086 4 10.6

Head loss due to bending:

30

Head loss due to straight Pipe:

31

Total Head loss 32

Pressure Drop: 33

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CHAPTER 4

Specification of Problem

4.1 Problem specification and Geometry Details:

Consider a steady state fluid flow through a pipe for different angle of pipe bending „α‟ shown in

the figure 1 and 2. The diameter and length of the channel are 2.5cm and 3.0 m respectively. The

inlet velocity range is 0.0401923m/s to 1.004809m/s, which is constant over the inlet cross

section. Pipe outlet is at pressure of 1 atm. As the fluid flows through the pipe, for a particular

Reynolds number its corresponding average velocity is applied to the inlet of the pipe. For the

case of isothermal, entropy generation is only due to viscous dissipation only. And for non-

isothermal cases entropy generation is due to both viscous dissipation and thermal dissipation. So

for each case its value is calculated and plotted and compared.

The computational domain of the cylindrical channels is represented in three dimensional (3D)

by a cylinder the geometry consists of a wall, inlet and outlet.

Figure 1 Geometry of the Pipe

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Figure 2 Geometry for α =600

4.2 Meshing of geometry:

Structured meshing method done in ANSYS Workbench was used for meshing the geometry.

Figure 3 Meshing for α= 600

bend pipe

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4.3 Physical Model

Based on the Reynolds number,

34

Where D is diameter of the pipe, is the density and is the viscosity of the fluid. Either

viscous laminar model or standard or k-ϵmodel is used for laminar and turbulent flow

respectively.

Table 2: Choice of model based on Reynolds number

Reynolds No.(Re) Flow Model

<2100 laminar

>2100 turbulent

4.4 Material Properties

The property of fluid is assumed to be constant for all the cases. Water based fluid properties:

Table 3: Property of fluid water

Kf (thermal conductivity, W/mK) 0.6

(viscosity, kg/ms) 0.001003

(density, kg/m3) 998.2

Cp(specific heat, Kj/kg K) 4.182

4.5 Boundary Conditions:

at the wall: Vr= 0, Vz=0

at inlet where Vx0 is a constant : Vr= 0, Vx= Vx0

at the outlet section providing a fully developed flow condition:

iso-thermal flow no thermal boundry condition

non-isothermal case, constant heat flux and constant wall temp are applied

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where Vr is the velocity in radial direction and Vx is the velocity in x direction

4.6 Method of solution:

There are two ways to solve the problem stated above: (i) analytical method and (ii) CFD

method. In the present study the entropy generation for different Reynolds number (for a given

inlet velocity of mass flow rate) to get a optimize value. And then to compare the analytical

values of the pressure drop with the CFD values of the pressure drop. The entropy generation

value for iso-thermal case is found out by the Equation 23 and for non-isothermal case the

entropy generation value is found out by Equation 20.

The CFD method follows the use of commercial software ANSYS Fluent 15.0 to solve the

problem. The specified solver in Fluent uses a pressure correction based iterative SIMPLE

algorithm with 2nd

order upwind scheme. The convergence criteria vary with flow rate.

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Chapter-5

Results and Discussion:

5.1 Validation

Pipes and pipe bendings contribute to major part of pressure drop as well as energy consumption.

So it is better to know pressure drop in a piping system.

Simulations have been done for the 1-weld mitered bend pipe for α equals to 450, 90

0 and 180

0

straight pipe and pressure drop over pipe is calculated for different Reynolds number. Fig. 4, 5

and 6 comparing the pressure drop in fig 4 it is clear that the 1800 shows the lowest pressure

drop. The angle 450 bending pipe shows the highest pressure drop for a particular Reynolds

number and the angle 900 bending have the pressure drop in between them.

Figure 4 Pressure drop versus Reynolds number plot for α =450

For very low flow velocity the pressure drop is nearly same for all the angle of bending but

difference in pressure drop among the bends increases as the velocity of the inlet fluid increases.

And then the calculated pressure drops are compared with values of the simulation pressure drop.

0

500

1000

1500

2000

2500

3000

3500

4000

0 5000 10000 15000 20000 25000 30000 35000

pre

ssu

re d

rop

(pas

cals

)

Reynolds number

cfd calculated value

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Figure 5 Pressure drop versus Reynolds number plot for α =900

In figure 4, 5 and 6 the values are compared and plotted for α =450, 90

0 and straight pipe 180

0

respectively. For low Reynolds number, the bending angles 450 and 90

0 the pressure drop values

are nearly the same. For moderate pressure drop we find a little deviation among them. And for

larger Reynolds number the difference grows larger. For the straight pipe up to moderate

Reynolds number the pressure drop variation is negligible and for higher Reynolds number the

difference in pressure drop gradually increases.

Table 4: Check For Mesh Independency by RMS error for straight pipe (α=1800)

Case Mesh Size RMSD NRMS %

1 72224 18.672 2.8754

2 121251 16.3185 2.513

3 218772 16.2563 2.503

Mesh independency are checked for α =1800 for three different mesh sizes as given in Table 4.

Pressure drop for each meshing is calculated for various Re and compared with corresponding

pressure drop found from empirical equation. For each case RMS error or deviation and

0

500

1000

1500

2000

2500

3000

0 5000 10000 15000 20000 25000 30000 35000

Pre

ssu

re d

rop

(pas

cals

)

Reynoldsd number

CFD value Calculated value

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NRMS% is calculated. Results show that for case 2 and case 3 NRMS values are nearly same.

Therefor further simulations are done using case 2 mesh size.

Figure 6 Pressure drop versus Reynolds number plot for α =180 0

5.2 prediction of entropy generation in pipefor isothermal boundary condition

The case is isothermal when the temperature is constant, i.e. the temperature of flowing fluid and

wall/ surrounding are kept same. In this case both the temperature of the water and wall are at

300 K. The entropy generation is only due to the viscous dissipation. Dissipation or viscous

dissipation is the source of energy supply to sustain turbulent flow. The turbulence dissipated

rapidly as kinetic energy is converted to internal energy by viscous shear stresses hence the

requirement of energy. But the temperature gradient developed due to this is very negligible.

The viscous dissipation depends upon two things. One is the velocity gradient in all three

directions (

) where is a function of velocity gradient and turbulent dissipation rate(

.Both

the viscous dissipation and turbulent dissipation rates increases as the flow rate increases, here

it‟s the Reynolds number.

Considering the bending angle α= 900, from the curve in Figure 10, the entropy generation value

increases as the Reynolds number increases.

0

100

200

300

400

500

600

700

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

Pre

ssu

re d

rop

(pas

cals

)

Reynolds number

cfd value calculated value

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(I-b) (II-b)

(III-b)

Figure 7 Contours of static pressure of α=900 at point b of the channel for (I) Reynolds no 1000,

(II) Reynolds no 5000 (III) for Reynolds no 20000

Figure 7 shows the contours of the pressure plot for Reynolds number 1000 and 5000. Pressure is

highest at the inlet and lowest at the outlet and the pressure gradually decreases from inlet to

outlet. As the Reynolds number increases the pressure drop also increases which can be clearly

seen from Fig. 7 (I) and (II), the static pressure value is highest for Reynolds number 20000

followed by Re5000 and then 1000. At the bending position towards the outer wall near point

(b), there is a small amount of increase and decrease in pressure, which is due to impact of the

fluid on the wall.

The kinetic energy of the moving fluid gets converted into pressure energy and the pressure

increases near the pipe bending. But at the same time at the opposite side of the wall there is a

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little higher pressure drop than the normal where the pressure energy is converted into kinetic

energy.

Figure.8 shows the velocity contour plot for α = 90° and Re = 1000, 5000 and 20000. For all the

contour plots it is observed that the velocity magnitude is less near the wall (nearly zero) and

highest at the center of the pipeline. As the wall is stationary frictional forces comes in to action

on the moving fluid. But as the flow rate increases (here Reynolds number) the magnitude of

velocity increases gradually from the wall to the center of the pipe.

(I-b) (II-b)

(III-b)

Figure 8 Velocity contours for α = 900 at point b of the channel for (I) for Reynolds number

1000, (II) for Reynolds number 5000, (III) for Reynolds no 20000

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At the bending of the pipe, towards the outer wall the velocity magnitude increases as the

potential energy of fluid transfers into the kinetic energy. Whereas at the other side of the wall of

the bend there is a decrease in velocity magnitude. At the inlet the velocity magnitude is little

higher than compared to other sections of the pipe. At the tip of bend, some vertex also forms

due to change in shape of the pipe. And the velocity of the fluid is nearly zero at that point. As

the Reynolds number increases the velocity magnitude also grows.

(I-b) (II-b)

(III-b)

Figure 9 Contours of local entropy generation values for α = 900at different points of the channel

for (a) Reynolds no 1000 (b) Reynolds no 5000 (c) Reynolds number 20000

Figure. 9 shows contour plots of local entropy generation. There are two portion of the pipe that

requires our attention. One is entrance region i.e. point (a) and another one is at the bending

towards the wall of the pipe i.e. point (b). In the entrance region the entropy generation is due to

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viscous force against the wall. In the entry region the velocity is higher and suddenly due to

opposing frictional forces velocity gradients gets developed. The entropy generation is the

function of square of velocity gradient (Eq. 23). While at the bending, it is due to a sudden

change in pressure drop as well as the velocity magnitude. Kinetic energy gets converted into

pressure energy and vice versa in that region. Due to these change in velocity magnitudes

velocity gradient develop, hence the entropy generation which is a function of square of velocity

gradient develops. Entropy generation rate near the wall increases sharply and decreases along

the pipe. And in the fluid due shear stresses the velocity gradient is developed, but entropy

generation values due to those are very small and can be neglected.

As the flow rate increases the magnitude of velocity increases accordingly velocity gradient also

increases. Along with it the turbulent dissipation rate also rises once the flow is turbulent. As the

entropy generation depends on both the viscous dissipation and turbulent dissipation rate for

isothermal case the entropy generation rate also rise as the Reynolds number increases. From the

figure.9 it is clear that the entropy generation values are more for higher Reynolds number.

Figure 10 Curve between Reynolds number and volumetric entropy generation for different α

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The entropy generation values for various bending angles are plotted against the Reynolds

number as shown in Figure.10 shows a constant increase in the entropy generation values.

Entropy generation value is highest for α = 450 bend pipe and lowest for α =180

0 bend pipe,

while others according to the increasing bending angle. But for Reynolds number near to 5000

the entropy generation value nearly same for all bending angles after that the difference in values

rises as the bending angle increases.

5.3 Non-isothermal boundary condition

5.3.1 Prediction of Entropy generation for Constant heat flux wall

Heat transfer is a thermodynamic irreversibility in all engineering devices. When heat is

transferred across a finite temperature difference, some capacity to do work is lost. And entropy

is generated.

Thermal flux or heat flux is the rate of heat energy transfer through a given surface per unit area.

It is a vector quantity. It is a derived quantity as it involves two quantities, heat transfer and area.

Heat flux is important in energy balance calculations. Normally it is used to maintain or establish

a steady state thermal gradient.

As the case is non-isothermal the temperature is not constant. Constant heat fluxes of different

values are applied on the wall of the pipe. And the entropy generation is studied. Now the

entropy generation is not only due to viscous dissipation but also for the thermal dissipation

values(

).

The amount of entropy generated increases the loss of available work. So the entropy generation

must be minimized to get optimum of the design of the thermal systems.

In the Fig. 11 contours of static temperature is plotted for bending angle 1200 at wall flux of

10000 W/m2 for Re=1000, 12000 and 30000. For Re 1000, Fig.11 (I) the temperature effect

distribution can be clearly seen. Temperature profile in the pipe changes continuously.

Temperature contours have high values in the surface region and as the distance from the pipe

wall towards the pipe center increases its magnitude of temperature contours reduces. The

temperature of the water gradually increases from the inlet towards the outlet.

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Wall boundary condition is uniform heat flux. So the temperature of the wall rises continuously

and heat is transferred form the wall to the fluid. Accordingly the temperature of the fluid is less

in the entrance region and higher at the exit region.

Comparing the contours of Figure 11 (I) (II) and (III) the static temperature values are higher for

low Reynolds number and low for higher Reynolds number. Residence time for low Reynolds

number is high in comparison to the residence time for high Reynolds number. So fluids flow for

low Reynolds number get more time for the transfer of heat into it.

(I-b) (II-b)

(III-b)

Figure 11 Temperature contour α = 1200 and wall flux = 10000W/m

2 at different points of the

channel for (I) Re= 1000, (II) Re=12000, (III) Re=30000

Figure12 Show the contour plots for thermal dissipation rates for α equals to 1200 and constant

wall flux of 10000 W/m2. As we know from equation (20) thermal dissipation is a function of

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temperature gradient. The more is the temperature variation the more will be the thermal

dissipation. As in case of Re=1000, Fig. 12(I) the temperature variation along the pipe is more so

the thermal dissipation value is higher. But for the Reynolds number 12000 and 30000 the

temperature values decreases as well as the magnitude of temperature gradients also decreases.

So the thermal dissipation rate is lower. So we can say that with rise in the Reynolds number the

temperature variation becomes less and the thermal dissipation value decreases.

(I-b)` (II-b)

(III-b)

Figure 12 Local thermal dissipation rate for bending angle 1200and wall flux 10000W/m

2at

different points of the channel for(I) Re= 1000, (II) Re=12000

From the velocity contour plot for pipe with α = 120° and Re = 12000. For the contour plot it is

observed that the velocity magnitude is less near the wall (nearly zero) and highest at the center

of the pipeline. As the wall is stationary frictional forces comes in to action on the moving fluid.

But as the Reynolds number (flow rate) the magnitude of velocity increases gradually from the

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wall to the center of the pipe. At the bending of the pipe, towards the outer wall the velocity

magnitude increases as the potential energy of fluid transfers into the kinetic energy. Whereas at

the other side/ inner side of the wall of the pipe bend there is a decreases in velocity magnitude

and rise in Static pressure magnitude. At the inlet the velocity magnitude is little higher than

compared to other sections of the pipe. At the tip of bend, some vertex also forms due to change

in shape of the pipe. And the velocity of the fluid is nearly zero at that point.

(I-b) (II-b)

(III-b)

Figure 13 Contours of Local viscous dissipation rate for α = 1200and wall flux 10000W/m

2 at

different points of the channel for (I) Re= 1000, (II) Re=12000, (III) Re=30000

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(I-b) (II-b)

(III-b)

Figure 14 Contours of Entropy Generation for α = 1200 and wall flux 10000W/m

2 at different

points of the channel for(I) Re= 1000, (II) Re=12000, (III) Re=30000

In the contours of entropy generation for low Reynolds number the entropy generation value is

higher because of the thermal dissipation rate. Viscous dissipation have very little effect on the

entropy generation for low Reynolds number. At the minimized entropy generation value as in

Figure (13) both the viscous dissipation term and thermal dissipation term have nearly the same

value. But for higher Reynolds number its value mostely depends upon the viscous dissipation.

The entropy generation value is the addition of viscous dissipation and thermal dissipation

values. At low Reynolds number the viscous dissipation values are low but as the Re value

increases entropy generation due to fictional losses increases. But at the thermal dissipation

values are high for low Re and its value decreases as the Re decreases. So entropy generation

minimization of occurs. In Figure 15 viscous dissipation, thermal dissipation and entopy

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generation for constant wall flux of 1000 W/m2 is plotted. And it can be clearly seen that entropy

generation passes value at first decreasesand after reaching a optimum value it again increases..

Figure 15 Curve between Reynolds number and volumetric entropy generation for α = 1200

angle of pipe bending at wall flux 10000W/m2.

Figure 16 Curve between Reynolds number and volumetric entropy generation for different α at

wall flux 1000W/m2

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Figure 17 Curve between Reynolds number and volumetric entropy generation for different α at

wall flux 5000W/m2

Figure 18 Curve between Reynolds number and volumetric entropy generation for different α at

wall flux 10000W/m2

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Figure 16, 17, 18 and 19 represents the plot between volumetric entropy generation and

Reynolds number for the entire bending angles considered at constant wall flux of 1000 W/m2,

5000 W/m2

10000 W/m2 and 20000 W/m

2. And each case it is found that the entropy generation

value decreases as the Reynolds number increases, then it reaches a minimum value and after

that point the entropy generation value increases. Entropy generation value is maximum for

α=45° and lowest for α = 180

° and as the α value increases the entropy generation value

decreases.

Here at the Reynolds number at which the entropy generation minimizes occur rises as the heat

flux is increases, Figure 20. Due to increase in heat flux the temperature in the wall increases so

heat is penetrated more to the water in comparison to lower heat flux. And thermal dissipation

value is increased for higher heat flux. But to reach the optimum value of the entropy generation

the frictional entropy generation value has to be increases. And this can be obtained only when

the Re value is increased.

Figure 19 Curve between Reynolds number and volumetric entropy generation for different α at

wall flux 20000W/m2

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Figure 20 Entropy Generation versus Re plot for different wall flux for α =1200

5.3.2 Prediction of Entropy generation for Constant wall Temperature

In the contours of static temperature the temperature effect distribution can be clearly seen.

Temperature profile in the pipe changes continuously. Temperature contours have high values in

the surface region and as the distance from the pipe wall towards the pipe center increases its

magnitude of temperature contours reduces. The temperature of the water gradually increases

from the inlet towards the outlet.

Wall boundary condition is uniform wall temperature. So due to temperature difference heat is

transferred form the wall to the fluid throughout the duct. Accordingly the temperature of the

fluid is less in the entrance region and higher at the exit region.

Comparing the contours of the static temperature values are higher for low Reynolds number and

low for higher Reynolds number. Residence time for low Reynolds number is high in

comparison to the residence time for high Reynolds number. So fluids flow for low Reynolds

number get more time for the transfer of heat into it.

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(I-b)

(II-b) (III-b)

Figure 21 Contours of Viscous dissipation for 600 bend wall temp=315K at different point-b of

the channel (I) Re= 1000, (II) Re=11000, (III) Re=20000

From the contour plots for thermal dissipation rates for bending angle 600

and constant wall

temperature 315K. As we know from equation (20) thermal dissipation is a function of

temperature gradient. The more is the temperature variation the more will be the thermal

dissipation. As in case of Re=1000, the temperature variation along the pipe is more so the

thermal dissipation value is higher. But for the Reynolds number 12000 and 30000 the

temperature values decreases as well as the magnitude of temperature gradients also decreases.

So the thermal dissipation rate is lower. So we can say that with rise in the Reynolds number the

temperature variation becomes less and the thermal dissipation value decreases.

The velocity magnitude is less near the wall (nearly zero) and highest at the center of the

pipeline. As the wall is stationary frictional forces comes in to action on the moving fluid. But as

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the Reynolds number (flow rate) the magnitude of velocity increases gradually from the wall to

the center of the pipe. At the bending of the pipe, towards the outer wall the velocity magnitude

increases as the potential energy of fluid transfers into the kinetic energy. Whereas at the other

(I-a)

(I-c)

Figure 22 Contours of Entropy Generation for α = 600 and wall temperature=315Kat different

points of the channel for (I) Re= 1000, (III) Re=20000

From the contours of local viscous dissipation rates (Figure 22). As it depends upon the function

and turbulent dissipation rate as in eq. 20 and 21. The viscous dissipation values are higher

at the entrance region, near the wall and at the pipe bending. At the entrance region the fluid

suddenly comes in contact with the stationary solid walls. And at the wall, which opposes the

fluid motion by frictional action also increases the viscous dissipation. Near the pipe bending due

to sudden change in geometry of the pipe, there is increase and decrease of pressure and velocity

term which adds to the viscous terms and thus to entropy generation. The value of viscous

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dissipation increases as the Reynolds number rises because velocity magnitude becomes higher

and due to frictional forces velocity gradient also increases.

In the contours of entropy generation for low Reynolds number the entropy generation value is

higher because of the thermal dissipation rate. Viscous dissipation have very little effect on the

entropy generation for low Reynolds number. At the minimized entropy generation value as in

both the viscous dissipation term and thermal dissipation term have nearly the same value. But

for higher Reynolds number its value mostly depends upon the viscous dissipation.

In the Figure 23 the entropy generation due to friction or viscous dissipation and thermal

dissipation are plotted verses Re. As the Re value increases the entropy generation due to

frictional loss also increases but the increment is very slow. By increasing the Re the frictional

losses increases and the value of velocity gradient accordingly increases. But for thermal

dissipation, the value at first increases with increase in Re up to a point then suddenly decreases

sharply. After that point it again decreases very slowly. This happens so because of the fact that

for small Reynolds number the residence time of water in the pipe is more in comparison to

higher Re number cases. Entropy generation is addition of those two values. So for low Re its

value is high and totally depends upon the thermal dissipation value but for lower Re it depends

upon the viscous dissipation.

Figure 23 Curve between Reynolds number and viscous dissipation and thermal dissipation for α

= 600 at wall Temp 315K

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Figure 24 Curve between Reynolds number and volumetric entropy generation for different α at

wall temp 305K

Figure 25 Curve between Reynolds number and volumetric entropy generation for different α at

wall temp 310K

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Figure 26 Curve between Reynolds number and volumetric entropy generation for different α at

wall temp 315K

Figure 27 Curve between Entropy Generation and Re plot for different wall temperature for α

=600

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Figure. 25, 26 and 27 represents the plot between volumetric entropy generation and Reynolds

number for the entire bending angles considered at constant wall flux of 305K, 310K and 315K.

And each case it is found that the entropy generation value at first increases up to Re 3000 then it

suddenly decreases and reaches a minimum value and after that point the entropy generation

value increases very slowly. For α = 450 bend has the highest entropy generation value and 180

0

has the lowest. As α value increases the entropy generation value decreases because with

increase in α the frictional value decreases.

In figure 28 entropy generation is plotted against Re for α = 600 for various wall temperature. As

the wall temperature is increased the thermal dissipation component of the total entropy

generation increases due to increase in temperature gradient within the fluid. As the temperature

at the wall increases more heat is penetrated through the wall intro the fluid. SO the temperature

gradient values also increases. But after reaching a maximum the entropy value decreases and

attains minima. After it the values remains nearly constant or increases very slowly. The entropy

generation rate is higher for higher wall temperature cases as the thermal component is greater

for larger wall temperature.

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Chapter-6

Conclusion:

Entropy generation is studied for the flow through a pipe with varying parameter α for three

different cases (I) iso-thermal case, (II) isoflux (III) constant wall temperature case.

1. As the Reynolds number increases the pressure drop also found to be increased

accordingly and it justifies all the pressure drop and velocity co-relations.

2. For isothermal (constant temperature throughout) case the entropy generation is due to

frictional irreversibility only. The entropy generation value increases with increase in Re. with

increase in Re the frictional forces increases and hence the velocity gradient.

3. The frictional component has two components. One is viscous dissipation and another

one is turbulence dissipation rate. Turbulence dissipation rate only occurs when the flow is

turbulent.

4. For lower α the frictional values are more. And as the value of α increases the entropy

generation values decreases

5. For non isothermal case (both isothermal wall temperature and iso-flux case) the entropy

generation value has two components. One is frictional irreversibility and other one is thermal

irreversibility.

6. For low Re numbers the viscous dissipation contribution towards the total entropy

generation rate is smaller in comparison to thermal dissipation component. Where as for large Re

cases the contribution of thermal and viscous irreversibility is just reverse.

7. The thermal dissipation value does not depend upon the high or low temperature. It

depends upon the temperature gradient. Total heat transfer is a function of the surface area and

length of the duct along with the residence time.

8. As the value of heat flux or wall temperature increases, point where minimum entropy

generation occur shifts towards higher Reynolds number.

9. Low is the angle α higher will be the entropy generation. And the value decreases as the α

increases for a particular flow rate. The frictional contribution is higher for lower α values.

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